2011/02/09 by Dorin Bucur, Bucur, Dorin, Ilaria Fragalà +3
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Point processes and geometric inequalities #math.OC
paper · pdf · doi:10.48550/arxiv.1102.1887
arxiv created 2011/02/09 · openalex publication_date 2011/02/09 · arxiv updated 2011/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by a long-standing conjecture of Polya and Szegö about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and indecomposability results, and we discuss their application to isoperimetriclike inequalities. As a byproduct of this approach we also obtain a quantitative version of the Kneser-Süss inequality. Finally, for a large class of functionals involving Dirichlet energies and the surface measure, we perform a local analysis of strictly convex portions of the boundary via second order shape derivatives. This allows in particular to exclude the presence of smooth regions with positive Gauss curvature in an optimal shape for Polya-Szegö problem.